Finite Group Databasebeta

← #20286346

Subgroups — #20286346

Order 6784.

498 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133246C211yesmaximal, characteristic, complemented
2#16811280C211yesmaximal, characteristic, complemented
3#16811267C211yesmaximal, characteristic, complemented
4#16811006C211yesmaximal, characteristic, complemented
5#16810881C211yesmaximal, characteristic, complemented
6#16810873C211yesmaximal, characteristic, complemented
7#16810843C211yesmaximal, characteristic, complemented
8#16810836C211yesmaximal, characteristic, complemented
9#16810785C211yesmaximal, characteristic, complemented
10#16810774C211yesmaximal, characteristic, complemented
11#16810765C211yesmaximal, characteristic, complemented
12#16810760C211yesmaximal, characteristic, complemented
13SD16×Dic53C211yesmaximal, characteristic, complemented
14#16810699C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C53C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×Q8×D53C2211yescharacteristic, complemented
18#13133246—24no—
19C2×Dic212C2211yescharacteristic, complemented
20C2×#11719925C2211yescharacteristic, complemented
21C2×#11719923C2211yescharacteristic
22C2×#11719923C2211yescharacteristic
23C2×#11719923C2211yescharacteristic, complemented
24C2×#11719912C2211yescharacteristic, complemented
25#13133204—21no—
26#13133204C2211yescharacteristic
27#13133204C2211yescharacteristic, complemented
28#13133198—21no—
29#13133196—21no—
30#13133196C2211yescharacteristic
31#13133196C2211yescharacteristic
32#13133189—21no—
33#13133188—21no—
34#13133188C2211yescharacteristic
35#13133188C2211yescharacteristic, complemented
36#13133186—21no—
37Q8×Dic53C2211yescharacteristic, complemented
38D4×Dic53C2211yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53C2211yescharacteristic
41C2×C22⋊Dic53—21no—
42#13133149C2211yescharacteristic, complemented
43#13133146C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133142C2211yescharacteristic, complemented
46#13133141C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133138C2211yescharacteristic, complemented
50C22⋊#88322C2211yescharacteristic
51#13133129C2211yescharacteristic, complemented
52C8×Dic53C2211yescharacteristic, complemented
53C2×#41×C53C2211yescharacteristic
54C22×D4×C53C2211yescharacteristic
55AGL(1,ℤ/8)×C53—24no—
56C2×SD16×C53C2211yescharacteristic
57C2×SD16×C53C2211yescharacteristic
58C2×D8×C53C2211yescharacteristic
59C2×D8×C53C2211yescharacteristic
60C2×M16×C53C2211yescharacteristic
61C24×C53—21no—
62#41×C53—22no—
63#41×C53—41no—
64C2×Q8×C53C2311yescharacteristic
65C2×D4×C53—22no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—41no—
69C2×D4×C53C2311yescharacteristic
70C2×D4×C53C2311yescharacteristic
71C2×D4×C53C2311yescharacteristic
72C22×C4×C53—21no—
73C22×C4×C53C2311yescharacteristic
74C2×C22⋊D53—21no—
75C2×C22⋊D53—21no—
76C2×C22⋊D53—21no—
77C2×C22⋊D53D411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—21no—
80Q8×D53—22no—
81Q8×D53—41no—
82#11719947—22no—
83#11719947—41no—
84C2×C4×D53—21no—
85C2×C4×D53C2311yescharacteristic
86C2×Dic106—21no—
87C2×Dic106C2311yescharacteristic
88C2×Dic106C2311yescharacteristic
89SD16×C53—22no—
90SD16×C53D414yescomplemented
91D8×C53—22no—
92D8×C53—22no—
93M16×C53—22no—
94C2×C8×C53C2311yescharacteristic
95C2×C8×C53C2311yescharacteristic
96C22⋊Dic53—21no—
97C22⋊Dic53—21no—
98C22⋊Dic53—21no—
99C22⋊Dic53—21no—
100C22⋊Dic53—21no—
101C22⋊Dic53—21no—
102C22⋊Dic53—21no—
103C22⋊Dic53—21no—
104C22⋊Dic53—21no—
105C22⋊Dic53D411yescharacteristic
106#11719925—22no—
107#11719923—22no—
108#11719923—41no—
109#11719923—41no—
110#11719921D411yescharacteristic
111C4⋊Dic53—21no—
112C4⋊Dic53C2311yescharacteristic
113C4⋊Dic53C2311yescharacteristic
114#11719919—21no—
115#11719919—21no—
116#11719919D411yescharacteristic
117C4×Dic53—21no—
118C4×Dic53C2311yescharacteristic
119C2×#88322C2311yescharacteristic
120C2×#88322C2311yescharacteristic
121Dic212—22no—
122#11719912—22no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—81no—
129C23×C53C2×D411yescharacteristic
130C22×D53C2×D411yescharacteristic
131Q8×C53—21no—
132Q8×C53C2×D412yes—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—21no—
136D4×C53—21no—
137D4×C53—21no—
138D4×C53—42no—
139D4×C53—42no—
140D4×C53C2×D412yes—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—42no—
145C2×C4×C53C2×D411yescharacteristic
146C2×C4×C53C2411yescharacteristic
147C22⋊D53—41no—
148C22⋊D53—42no—
149C22⋊D53—42no—
150C22⋊D53—81no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—42no—
156C2×Dic53—81no—
157C2×Dic53C2×D411yescharacteristic
158C2×Dic53C2×D411yescharacteristic
159C2×Dic53C2×D411yescharacteristic
160C4×D53—21no—
161C4×D53—21no—
162C4×D53—42no—
163Dic106—21no—
164Dic106—21no—
165Dic106—21no—
166Dic106—21no—
167Dic106—42no—
168C8×C53—21no—
169C8×C53C2×D412yes—
170#88322—21no—
171#88322—21no—
172C22×C53—22no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—41no—
184C22×C53—84no—
185C22×C53C22×D411yescharacteristic
186C2×D53—21no—
187C2×D53—21no—
188C2×D53—21no—
189C4×C53—22no—
190C4×C53—21no—
191C4×C53C22×D411yescharacteristic
192C4×C53H(5,𝔽2)11yescharacteristic
193Dic53—21no—
194Dic53—21no—
195Dic53—21no—
196Dic53—41no—
197Dic53—41no—
198Dic53—41no—
199Dic53—41no—
200#3275—531nomaximal, Sylow
201C2×C53—41no—
202C2×C53—42no—
203C2×C53—81no—
204C2×C53—81no—
205C2×C53D4211yescharacteristic
206C2×C53#57511yescharacteristic
207C2×C53#57711yescharacteristic
208D53—41no—
209C2×#137—531no—
210C2×AGL(1,ℤ/8)D5311yescharacteristic, complemented
211#548—531no—
212#546—531no—
213#495—531no—
214#470—531no—
215#469—531no—
216#464—531no—
217#463—531no—
218#452—531no—
219#450—531no—
220#449—531no—
221#448—531no—
222C4×SD16—531no—
223#436—531no—
224C53#327511yescharacteristic, complemented, Sylow
225C2×#41—531no—
226C2×#41C2×D5311yescharacteristic
227C22×Q8—531no—
228C22×D4C2×D5311yescharacteristic
229#137—1064no—
230AGL(1,ℤ/8)—24no—
231C2×Q16—531no—
232C2×Q16—531no—
233C2×SD16—531no—
234C2×SD16—531no—
235C2×SD16—531no—
236C2×SD16C2×D5311yescharacteristic
237C2×SD16C2×D5311yescharacteristic
238C2×D8C2×D5311yescharacteristic
239C2×D8C2×D5311yescharacteristic
240C2×M16—531no—
241C2×M16C2×D5311yescharacteristic
242#124—531no—
243#124—531no—
244#124—1061no—
245#123—1061no—
246C22⋊Q8—531no—
247C22⋊Q8—531no—
248C22⋊Q8—1061no—
249#121—531no—
250#121—531no—
251#121—1061no—
252#121—1061no—
253#120—1061no—
254C4×Q8—531no—
255C4×D4—531no—
256C4×D4—531no—
257C4×D4—1061no—
258C2×C22⋊C4—1061no—
259#106—531no—
260C4⋊C8—531no—
261#103—531no—
262#103—531no—
263#103—531no—
264#102—531no—
265#102—531no—
266#102—531no—
267C22⋊C8—531no—
268C22⋊C8—531no—
269C4×C8—531no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—1062no—
274#41—2121no—
275C2×Q8—531no—
276C2×Q8—531no—
277C2×Q8—1061no—
278C2×Q8—1062no—
279C2×Q8—2121no—
280C2×Q8C22×D5311yescharacteristic
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—531no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4C22×D5311yescharacteristic
290C2×D4C22×D5311yescharacteristic
291C2×D4C22×D5311yescharacteristic
292C22×C4—21no—
293C22×C4—531no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—1061no—
297C22×C4C22×D5311yescharacteristic
298Q16—1062no—
299Q16—1062no—
300SD16—22no—
301SD16—1062no—
302SD16—2121no—
303SD16—2121no—
304SD16C22⋊D5314yescomplemented
305D8—22no—
306D8—22no—
307M16—22no—
308M16—1062no—
309C2×C8—531no—
310C2×C8—531no—
311C2×C8C22×D5311yescharacteristic
312C2×C8C22×D5311yescharacteristic
313C4⋊C4—531no—
314C4⋊C4—531no—
315C4⋊C4—531no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—1061no—
319C22⋊C4—531no—
320C22⋊C4—531no—
321C22⋊C4—1061no—
322C22⋊C4—1061no—
323C22⋊C4—1061no—
324C22⋊C4—1061no—
325C22⋊C4—1061no—
326C22⋊C4—1061no—
327C22⋊C4—1061no—
328C22⋊C4—1061no—
329C22⋊C4—1061no—
330C42—531no—
331C42—1061no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—531no—
339C23D4×D5311yescharacteristic
340Q8—21no—
341Q8—1061no—
342Q8—1061no—
343Q8—1061no—
344Q8—1061no—
345Q8—2122no—
346Q8C2×C22⋊D5312yes—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—42no—
353D4—42no—
354D4—4241no—
355D4—2121no—
356D4—2122no—
357D4—2122no—
358D4C2×C22⋊D5312yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—531no—
364C2×C4—531no—
365C2×C4—531no—
366C2×C4—1061no—
367C2×C4—1061no—
368C2×C4—1061no—
369C2×C4—1061no—
370C2×C4—1061no—
371C2×C4—1061no—
372C2×C4—4241no—
373C2×C4—2122no—
374C2×C4—2122no—
375C2×C4D4×D5311yescharacteristic
376C2×C4C23×D5311yescharacteristic
377C8—21no—
378C8—1061no—
379C8—1061no—
380C8C2×C22⋊D5312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—1061no—
395C22—1061no—
396C22—1061no—
397C22C2×D4×D5311yescharacteristic
398C4—21no—
399C4—22no—
400C4—1061no—
401C4—1061no—
402C4—1061no—
403C4—2121no—
404C4—2121no—
405C4—2121no—
406C4—2121no—
407C4C22×C22⋊D5311yescharacteristic
408C4#1313326211yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—81no—
413C2—2121no—
414C2D4×C22⋊D5311yescharacteristic
415C2#1681141911yescharacteristic
416C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×AGL(1,ℤ/8)×C53C2
socleC22×C53C22×D4
hypercenterC2×AGL(1,ℤ/8)D53
hypocenterC53#3275
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#13133246C2semidirect
2. #16811280C2semidirect
3. #16811267C2semidirect
4. #16811006C2semidirect
5. #16810881C2semidirect
6. #16810873C2semidirect
7. #16810843C2semidirect
8. #16810836C2semidirect
9. #16810785C2semidirect
10. #16810774C2semidirect
11. #16810765C2semidirect
12. #16810760C2semidirect
13. SD16×Dic53C2semidirect
14. #16810699C2semidirect
15. C2×AGL(1,ℤ/8)×C53C2semidirect
17. C2×Q8×D53C22semidirect
19. C2×Dic212C22semidirect
20. C2×#11719925C22semidirect
23. C2×#11719923C22semidirect
24. C2×#11719912C22semidirect
27. #13133204C22semidirect
35. #13133188C22semidirect
37. Q8×Dic53C22semidirect
42. #13133149C22semidirect
43. #13133146C22semidirect
44. #13133142C22semidirect
45. #13133142C22semidirect
46. #13133141C22semidirect
49. #13133138C22semidirect
51. #13133129C22semidirect
52. C8×Dic53C22semidirect
90. SD16×C53D4semidirect
210. C2×AGL(1,ℤ/8)D53semidirect
224. C53#3275semidirect
304. SD16C22⋊D53semidirect